90.3.21 problem 21

Internal problem ID [25085]
Book : Ordinary Differential Equations. By William Adkins and Mark G Davidson. Springer. NY. 2010. ISBN 978-1-4614-3617-1
Section : Chapter 1. First order differential equations. Exercises at page 41
Problem number : 21
Date solved : Thursday, October 02, 2025 at 11:49:41 PM
CAS classification : [_separable]

\begin{align*} y+1+\left (-1+y\right ) \left (t^{2}+1\right ) y^{\prime }&=0 \end{align*}

✓ Maple. Time used: 0.010 (sec). Leaf size: 21

ode:=y(t)+1+(y(t)-1)*(t^2+1)*diff(y(t),t) = 0; 
dsolve(ode,y(t), singsol=all);
 

\[ y = -2 \operatorname {LambertW}\left (-\frac {{\mathrm e}^{\frac {\arctan \left (t \right )}{2}-\frac {1}{2}+\frac {c_1}{2}}}{2}\right )-1 \]

✓ Mathematica. Time used: 6.649 (sec). Leaf size: 60

ode=y[t]+1+(y[t]-1)*(1+t^2)*D[y[t],{t,1}] ==0; 
ic={}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to -1-2 W\left (-\frac {1}{2} \sqrt {e^{\arctan (t)-1-c_1}}\right )\\ y(t)&\to -1-2 W\left (\frac {1}{2} \sqrt {e^{\arctan (t)-1-c_1}}\right )\\ y(t)&\to -1 \end{align*}

✓ Sympy. Time used: 0.519 (sec). Leaf size: 53

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq((t**2 + 1)*(y(t) - 1)*Derivative(y(t), t) + y(t) + 1,0) 
ics = {} 
dsolve(ode,func=y(t),ics=ics)
 

\[ \left [ y{\left (t \right )} = - 2 W\left (- \frac {\sqrt {C_{1} e^{\operatorname {atan}{\left (t \right )}}}}{2 e^{\frac {1}{2}}}\right ) - 1, \ y{\left (t \right )} = - 2 W\left (\frac {\sqrt {C_{1} e^{\operatorname {atan}{\left (t \right )}}}}{2 e^{\frac {1}{2}}}\right ) - 1\right ] \]